Optimal. Leaf size=237 \[ \frac{2 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{b d+2 c d x}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right ),-1\right )}{c d^{3/2} \sqrt [4]{b^2-4 a c} \sqrt{a+b x+c x^2}}-\frac{2 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{c d^{3/2} \sqrt [4]{b^2-4 a c} \sqrt{a+b x+c x^2}}+\frac{4 \sqrt{a+b x+c x^2}}{d \left (b^2-4 a c\right ) \sqrt{b d+2 c d x}} \]
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Rubi [A] time = 0.209, antiderivative size = 237, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {693, 691, 690, 307, 221, 1199, 424} \[ \frac{2 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{c d^{3/2} \sqrt [4]{b^2-4 a c} \sqrt{a+b x+c x^2}}-\frac{2 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{c d^{3/2} \sqrt [4]{b^2-4 a c} \sqrt{a+b x+c x^2}}+\frac{4 \sqrt{a+b x+c x^2}}{d \left (b^2-4 a c\right ) \sqrt{b d+2 c d x}} \]
Antiderivative was successfully verified.
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Rule 693
Rule 691
Rule 690
Rule 307
Rule 221
Rule 1199
Rule 424
Rubi steps
\begin{align*} \int \frac{1}{(b d+2 c d x)^{3/2} \sqrt{a+b x+c x^2}} \, dx &=\frac{4 \sqrt{a+b x+c x^2}}{\left (b^2-4 a c\right ) d \sqrt{b d+2 c d x}}-\frac{\int \frac{\sqrt{b d+2 c d x}}{\sqrt{a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right ) d^2}\\ &=\frac{4 \sqrt{a+b x+c x^2}}{\left (b^2-4 a c\right ) d \sqrt{b d+2 c d x}}-\frac{\sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \int \frac{\sqrt{b d+2 c d x}}{\sqrt{-\frac{a c}{b^2-4 a c}-\frac{b c x}{b^2-4 a c}-\frac{c^2 x^2}{b^2-4 a c}}} \, dx}{\left (b^2-4 a c\right ) d^2 \sqrt{a+b x+c x^2}}\\ &=\frac{4 \sqrt{a+b x+c x^2}}{\left (b^2-4 a c\right ) d \sqrt{b d+2 c d x}}-\frac{\left (2 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1-\frac{x^4}{\left (b^2-4 a c\right ) d^2}}} \, dx,x,\sqrt{b d+2 c d x}\right )}{c \left (b^2-4 a c\right ) d^3 \sqrt{a+b x+c x^2}}\\ &=\frac{4 \sqrt{a+b x+c x^2}}{\left (b^2-4 a c\right ) d \sqrt{b d+2 c d x}}+\frac{\left (2 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^4}{\left (b^2-4 a c\right ) d^2}}} \, dx,x,\sqrt{b d+2 c d x}\right )}{c \sqrt{b^2-4 a c} d^2 \sqrt{a+b x+c x^2}}-\frac{\left (2 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1+\frac{x^2}{\sqrt{b^2-4 a c} d}}{\sqrt{1-\frac{x^4}{\left (b^2-4 a c\right ) d^2}}} \, dx,x,\sqrt{b d+2 c d x}\right )}{c \sqrt{b^2-4 a c} d^2 \sqrt{a+b x+c x^2}}\\ &=\frac{4 \sqrt{a+b x+c x^2}}{\left (b^2-4 a c\right ) d \sqrt{b d+2 c d x}}+\frac{2 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{c \sqrt [4]{b^2-4 a c} d^{3/2} \sqrt{a+b x+c x^2}}-\frac{\left (2 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x^2}{\sqrt{b^2-4 a c} d}}}{\sqrt{1-\frac{x^2}{\sqrt{b^2-4 a c} d}}} \, dx,x,\sqrt{b d+2 c d x}\right )}{c \sqrt{b^2-4 a c} d^2 \sqrt{a+b x+c x^2}}\\ &=\frac{4 \sqrt{a+b x+c x^2}}{\left (b^2-4 a c\right ) d \sqrt{b d+2 c d x}}-\frac{2 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{c \sqrt [4]{b^2-4 a c} d^{3/2} \sqrt{a+b x+c x^2}}+\frac{2 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{c \sqrt [4]{b^2-4 a c} d^{3/2} \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 0.0442038, size = 89, normalized size = 0.38 \[ -\frac{2 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{c d \sqrt{a+x (b+c x)} \sqrt{d (b+2 c x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.251, size = 336, normalized size = 1.4 \begin{align*}{\frac{1}{c{d}^{2} \left ( 2\,{c}^{2}{x}^{3}+3\,bc{x}^{2}+2\,acx+{b}^{2}x+ab \right ) \left ( 4\,ac-{b}^{2} \right ) }\sqrt{d \left ( 2\,cx+b \right ) }\sqrt{c{x}^{2}+bx+a} \left ( 4\,{\it EllipticE} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) ac\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}-{\it EllipticE} \left ({\frac{\sqrt{2}}{2}\sqrt{{ \left ( b+2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}},\sqrt{2} \right ){b}^{2}\sqrt{{ \left ( b+2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}\sqrt{-{(2\,cx+b){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}\sqrt{{ \left ( -b-2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}-4\,{c}^{2}{x}^{2}-4\,bcx-4\,ac \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} \sqrt{c x^{2} + b x + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{2 \, c d x + b d} \sqrt{c x^{2} + b x + a}}{4 \, c^{3} d^{2} x^{4} + 8 \, b c^{2} d^{2} x^{3} + a b^{2} d^{2} +{\left (5 \, b^{2} c + 4 \, a c^{2}\right )} d^{2} x^{2} +{\left (b^{3} + 4 \, a b c\right )} d^{2} x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (d \left (b + 2 c x\right )\right )^{\frac{3}{2}} \sqrt{a + b x + c x^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} \sqrt{c x^{2} + b x + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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